Interchanging Rows Of Matrix Changes Sign Of Determinants!  TOPIC_SOLVED

Linear spaces and subspaces, linear transformations, bases, etc.

Interchanging Rows Of Matrix Changes Sign Of Determinants!

Postby RedPrince007 on Tue Oct 08, 2013 5:00 pm

Hello Everyone..!!
Now A Days I Am Learning About Matrix And Determinants And I Confused On One Properties Of Determinants Which Is: Interchanging Two Rows/Columns Of A Determinant Changes The Sign Of The Determinant!

My Question Is What Is The Logic(Reason) That -ve Sign Is Places Outside The Determinants While Interchanging Rows/Columns But No Sign Is Places Outsides In Gaussian Elimination (OR More Specific In Matrix)

I Don't Understand The Logic Behind This! I Google It A Lot But Found No Answer Under The Scope Of My Knowledge.! Can Anybody Please Explain Why We Do This..!! Thanks In Advance For Answer! And Sorry In Advance If I Am Not Specific..!!
RedPrince007
 
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Re: Interchanging Rows Of Matrix Changes Sign Of Determinant

Postby buddy on Tue Oct 08, 2013 6:45 pm

Try doing a 2x2 and see what happens.

Code: Select all
original:

| -3  2 |
|  1  5 | = (-3)(5) - (1)(2) = -15 - 2 = -17

swapped:

|  2 -3 |
|  5  1 | = (2)(1) - (5)(-3) = 2 - (-15) = 17


Since any determinant can be done with 2x2s using minors and cofactors this shows how the sign changes: all the 2x2s flipped signs, so the whole thing does too. You probably could prove it with induction.
buddy
 
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Re: Interchanging Rows Of Matrix Changes Sign Of Determinant

Postby RedPrince007 on Tue Oct 08, 2013 7:14 pm

Why We Place -ive Sign Outside Determinant It Is Clear To Me Now But My Second Question Is That We Place -ive Sign When We Interchange Rows Of Determinant But Why We Do Not Place -ive Sign When We Interchange Rows In Gaussian Elimination i.e., While Performing Row Operations!!
RedPrince007
 
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Re: Interchanging Rows Of Matrix Changes Sign Of Determinant  TOPIC_SOLVED

Postby buddy on Tue Oct 08, 2013 10:05 pm

Its a different process for row ops on a matrix. Theres not "-1 to a power" for the matrix, only for the determinant. If you do something different & use a different process, you get different answers. Determinants aren't matrixes.
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